#
Existence and Uniqueness of Solutions: Example 1

**Example:** Suppose the differential equation satisfies the Existence and Uniqueness Theorem
for all values of *y* and *t*. Suppose and
are two solutions to this differential equation.

**1.**
- What can you say about the behavior of the solution of the
solution
*y*(*t*) satisfying the initial condition *y*(0)=1?

Hint: Draw the two solutions and .
**2.**
- Address the behavior of
*y*(*t*) as *t* approaches ,
and as *t* approaches .

## Answer:

**1.**
- First let us draw the graphs of and .

Since we have , we deduce from the
Existence and Uniqueness Theorem that for all *t*, we have

In particular, *y*(*t*) has the line *y*=*t* as an oblique asymptote
which answers the second question.

We cannot predict that *y*(*t*) is an increasing function.

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[Differential Equations]
[Slope Field]
**** **
[Geometry]
[Algebra]
[Trigonometry ]
[Calculus]
[Complex Variables]
[Matrix Algebra]

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*
Author:
Helmut Knaust *

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